| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Show definite integral equals specific value (requiring partial fractions or complex algebra) |
| Difficulty | Standard +0.8 This question requires executing a non-trivial substitution (involving a square root), correctly transforming limits and the integrand, then applying partial fractions and logarithmic integration. While the substitution is given, students must handle the algebraic manipulation carefully and complete multiple technical steps without error, making it moderately challenging but within reach for well-prepared A-level students. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| State or imply \(2u\,du = -dx\), or equivalent | B1 | |
| Substitute for \(x\) and \(dx\) throughout | M1 | |
| Obtain integrand \(\frac{-10u}{6-u^2+u}\), or equivalent | A1 | |
| Show correct working to justify the change in limits and obtain the given answer correctly | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| State or imply the form of fractions \(\frac{A}{3-u} + \frac{B}{2+u}\) and use a relevant method to find \(A\) or \(B\) | M1 | |
| Obtain \(A = 6\) and \(B = -4\) | A1 | |
| Integrate and obtain \(-6\ln(3-u) - 4\ln(2+u)\), or equivalent | A1⬇ + A1⬇ | |
| Substitute limits correctly in an integral of the form \(a\ln(3-u) + b\ln(2+u)\) | M1 | |
| Obtain the given answer correctly having shown sufficient working | A1 | [6] |
**(i)**
| State or imply $2u\,du = -dx$, or equivalent | B1 | |
| Substitute for $x$ and $dx$ throughout | M1 | |
| Obtain integrand $\frac{-10u}{6-u^2+u}$, or equivalent | A1 | |
| Show correct working to justify the change in limits and obtain the given answer correctly | A1 | [4] |
**(ii)**
| State or imply the form of fractions $\frac{A}{3-u} + \frac{B}{2+u}$ and use a relevant method to find $A$ or $B$ | M1 | |
| Obtain $A = 6$ and $B = -4$ | A1 | |
| Integrate and obtain $-6\ln(3-u) - 4\ln(2+u)$, or equivalent | A1⬇ + A1⬇ | |
| Substitute limits correctly in an integral of the form $a\ln(3-u) + b\ln(2+u)$ | M1 | |
| Obtain the given answer correctly having shown sufficient working | A1 | [6] |
**Note:** [The f.t. is on $A$ and $B$.]
---8 Let $I = \int _ { 2 } ^ { 5 } \frac { 5 } { x + \sqrt { } ( 6 - x ) } \mathrm { d } x$.\\
(i) Using the substitution $u = \sqrt { } ( 6 - x )$, show that
$$I = \int _ { 1 } ^ { 2 } \frac { 10 u } { ( 3 - u ) ( 2 + u ) } \mathrm { d } u$$
(ii) Hence show that $I = 2 \ln \left( \frac { 9 } { 2 } \right)$.
\hfill \mbox{\textit{CAIE P3 2012 Q8 [10]}}